Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use ...

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Extensions over finite fields

Let $K$ and $L$ be extensions of a finite field $F$ of degrees $n$ and $m$,respectively. How do I show that $KL$ has degree $\mathrm{lcm}(m,n)$ over $F$ and $K\cap F$ has degree $\gcd(m,n)$ over $F$? ...
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Galois group of finite extensions

Given a finite extension $E/k$ of a field $k$, how do I prove the following? $$|\operatorname{Gal}(E/k)| \text{ divides } [E:k].$$ Thanks.
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96 views

Why is the trace map from a finite extension of $\mathbb{Q}_p$ continuous?

Let $k$ be a finite extension of $\mathbb{Q}_{p}$. Why is $tr_{k/\mathbb{Q}_{p}}$ a continuous map from $k$ onto $\mathbb{Q}_{p}$?
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Polynomial interpolation of the residues of a rational function

Let $g(z) = a\prod_{i=1}^N (z-\lambda_i) \in \mathbb{Q}[z]$ be square-free. At each root $\lambda_i \in \mathbb{C}$, let $r_i$ denote the residue $\mathrm{Res}_{\lambda_i} 1/g(z)$. Let $I_g(z)$ ...
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Let K/F be a finite extension, given a polynomial in K[x] find another so that their product is in F[x]

Let $K$ be a finite extension of a field $F$, and let $f(x)$ be in $K[x]$. Prove that there is a nonzero polynomial $g(x)$ in $K[x]$ such that $f(x)g(x)$ is in $F[x]$. Should I do this by induction ...
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Corestriction in Galois cohomology (Serre, Corps Locaux)

I have a question about Chapter XIV of "Corps Locaux" by Serre. Sketch of the situation: let $K$ be a field with separable closure $\overline{K}$. Let $\Gamma_K$ be the absolute Galois group. The ...
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698 views

Generator and char. polynomial for a binary Galois Field produced by an external-XOR LFSR

My question is regarding LFSRs (Linear Feedback Shift Registers), and the binary Galois Field produced by them (also commonly termed GF($2^n$) ). I understand that a given n-bit LFSR produces a ...
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414 views

Reference request: Abel or Ruffini's proof of the Abel-Ruffini theorem

The standard proof of the Abel-Ruffini theorem that people learn is based on Galois theory and the notion of a solvable group, but my understanding is that the original proof predates Galois theory. ...
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If $L/k$ is Galois and $k\subseteq K$ is any field extension, is $LK$ Galois over $L$

There is a theorem in Lang which says that if $L/k$ is Galois and $k\subseteq K$ is any field extension, if $L,K$ are subfields of a larger field, then $LK$ is Galois over $K$. I was wondering if $LK$ ...
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How to show that a subfield of a Galois extension with Galois group $S_n$ has only trivial automorphisms.

Let $K$ be the splitting field of $f(x)\in\mathbb{Q}[x]$ over $\mathbb{Q}$, of degree $n$, and suppose that $\operatorname{Gal}(K/\mathbb{Q})=S_n$. It is easy to show that his implies that $f$ is ...
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How to show that a prime degree separable field extension containing a nontrivial conjugate of a primitive element is Galois and cyclic

Let $F$ be a field and let $E/F$ be a separable extension, with $[E:F]=p$, a prime. Given a primitive $\alpha_1\in E$ with $F(\alpha_1)=E$. Let $\alpha_2\neq \alpha_1$ denote one of the $p$ conjugates ...
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263 views

Is there something like Cardano's method for a SOLVABLE quintic.

So there is no quadratic formula equivalent for a GENERAL fifth degree equation, but is there an equivalent formula for a SOLVABLE fifth degree equation.
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If a polynomial has only rational roots does that automatically mean it is solvable?

Note that I am talking about rational roots not rational coefficients. I know that Galois theory can tell you but I want to know if knowing whether all the roots of a polynomial are rational can also ...
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137 views

Map from absolute Galois group of a global field to that of one of its completions

Let $K$ be a global number/function field, and let $v$ be a place of $K$. How to construct an explicit map from $G(\overline{K}/K)\rightarrow G(\overline{K}_{v}/K_v)$?
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Compositum of fields preserve finiteness and Galois-ity

Let $E/F$ be a finite Galois extension. Let $K$ be a function field of transcendence degree one over $F$. Let $KE$ be the compositum of $K$ and $E$. Why is $KE/K$ also finite and Galois? Also, why is ...
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Are these two notions of Galois morphism the same

Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$. Are the following two conditions equivalent? The function field extension $K(Y)\subset ...
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144 views

Solving an equation in a cyclotomic field

Let $\zeta = e^{2\pi i/110}$, and set $K = \mathbb{Q}(\zeta)$. There is an $\alpha$ in $\mathbb{Q}(\zeta^{11})$ of absolute value 1, which I'm trying to find. Consider $\sigma$, the Galois ...
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190 views

Does the sum generate the field compositum when degrees are prime?

Inspired by Finding the minimal polynomial of $\sqrt 2 + \sqrt[3] 2$ over $\mathbb Q$. Suppose $k$ is a field and $\alpha,\beta$ are algebraic over $k$ such that $[k(\alpha):k]=p$ and ...
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534 views

What are the quadratic extensions of $\mathbb{Q}_2$?

How do you classify the non-squares in $\mathbb{Q}_2$? I've tried writing down expansions for "odd" numbers in $\mathbb{Z}_2$, but unlike in $\mathbb{Z}_p$, the n$^{th}$ term in the expansion is not ...
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Is $\operatorname{Gal}(\mathbb{Q}_p^{un})\cong \hat{\mathbb{Z}}$?

Is the absolute Galois group of $\mathbb{Q}_p^{un}$ the profinite completion of $\mathbb{Z}$? I was never quite sure... In similar cases, it is true. Namely, $\mathbb{C}((t))$ does have absolute ...
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565 views

A Galois Group Problem

I couldn't figure out a proof of the following statement while I'm reading the book "Fields and Galois Theory" by J. Milne. Let $A$ to be a UFD and let $P$ be a prime ideal of $A$, and let ...
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Minimal polynomial of intermediate extensions under Galois extensions.

Let $K$ be a Galois extension of $F$, and let $a\in K$. Let $n=[K:F]$, $r=[F(a):F]$, and $H=\mathrm{Gal}(K/F(a))$. Let $z_1, z_2,\ldots,z_r$ be left coset representatives of $H$ in $G$. Show that ...
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Example of finite extension that is not prime with no proper intermediate extension. [duplicate]

Possible Duplicate: If a normal $K/F$ has no intermediate extensions, then $\[K : F\]$ is prime I want an example of: K is a finite extension of F.There is no proper intermediate extensions ...
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344 views

Example of Galois extension whose Galois group is $A_4$

Can anyone give me an example of a finite Galois extension whose Galois group is $A_4$?
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171 views

Fixed field of $G=\operatorname{Gal}(k(x)/k)$ in $k$

Let $k$ be a field and $k(x)$ be rational field of k and $G=\operatorname{Gal}(k(x)/k)$ $$k(x)^G=\{a \in k(x): \sigma(a)=a\text{ for all }\sigma \in G \}$$ How can we calculate $k(x)^G$?
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483 views

If a normal $K/F$ has no intermediate extensions, then $[K : F]$ is prime

Let $K$ be a finite normal extension of $F$ such that there are no proper intermediate extensions of $K/F$. Show that $[K:F]$ is prime. Give a conterexample if $K$ is not normal over $F$.
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Galois over Galois

I am working on this exercise: If $E$ is an intermediate field of an extension $F/K$ of fields. Suppose $F/E$ and $E/K$ are Galois extensions, and every $\sigma\in Gal(E/K)$ is extendible to an ...
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Finding an $F$-automorphism, having specified the images of two elements

Let $M$ be a normal extension of $F$. Suppose that $a_1, a_2$ are in $M$ and are the roots of the minimal polynomial of $a_1$ over $F$, and $b_1,b_2$ are the roots of minimal polynomial of $b_1$ over ...
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How to prove that the Galois group of a normal extension transitively permutes the factors of an irreducible polynomial?

How to do the following problem? Let $K$ be a normal extension of $F$, and let $f(x)\in F[x]$ be an irreducible polynomial over $F$. Let $g(x)$ and $p(x)$ be monic irreducible factors of $f(x)$ ...
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59 views

Number of solutions to a 0-1 non-linear integer program

When all the input variables $a_i$ are restricted to $\{0,1\}$, how does one compute the number of solutions to equations like $$ a_1a_4a_2 + a_1a_5a_3 + a_4a_6a_3 = c $$ where $c$ is a ...
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Inverse Limits: Isomorphism between Gal$(\mathbb{Q}(\cup_{n \geq 1}\mu_n)/\mathbb{Q})$ and $\varprojlim (\mathbb{Z}/n\mathbb{Z})^\times$

I'm trying to prove that $\operatorname{Gal}(\mathbb{Q}(\cup_{n \geq 1}\mu_n)/\mathbb{Q}) \cong \widehat{\mathbb{Z}}^\times = (\varprojlim (\mathbb{Z}/n\mathbb{Z}))^\times$, where $\varprojlim$ ...
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Complex conjugation in the Galois group of a polynomial

Suppose $P$ is an irreducible polynomial in $\mathbb Q[X]$, with exactly two non-real roots. Then we know these roots must be complex conjugates. Why must complex conjugation be an element of ...
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Galois group of a cubic over the function field $\mathbb Q(y)$

Given the polynomial $x^{3} - 3yx + 2 = 0$ with coefficients in $\mathbb Q(y)$ where $y$ is an indeterminate, which has discriminant $108(y^3 - 1)$, what is the Galois group of this polynomial? ...
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Galois group of $X^4 + 4X^2 + 2$ over $\mathbb Q$.

I'd like to calculate the Galois group of the polynomial $f = X^4 + 4X^2 + 2$ over $\mathbb Q$. My thoughts so far: By Eisenstein, $f$ is irreducible over $\mathbb Q$. So $\mathrm{Gal}(f)$ must be ...
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Is there a quick way to compute the degree of the splitting field of $x^3+x+1$ over $\mathbb{Q}$?

Is there a way to find the degree of the splitting field of $x^3+x+1$ over $\mathbb{Q}$? Just analyzing the roots shows that the polynomial is separable, so I suppose the splitting field would be a ...
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280 views

Calculating $\prod (\omega^j - \omega^k)$ where $\omega^n=1$.

Let $1, \omega, \dots, \omega^{n-1}$ be the roots of the equation $z^n-1=0$, so that the roots form a regular $n$-gon in the complex plane. I would like to calculate $$ \prod_{j \ne k} (\omega^j - ...
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553 views

Examples of Galois connections?

On TWF week 201, J. Baez explains the basics of Galois theory, and say at the end : But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If ...
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Original works of great mathematician Évariste Galois

Through this question I wanted to know the original works of Galois. When I was reading Galois theory ( since from last month ) , I have been seeing one common line in every book, whose essence ...
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Derivations and separability of field extensions

This is written on page 4 of James E. Humphreys' Linear Algebraic Groups: A derivation $\delta: E \rightarrow L$ ($E$ a field, $L$ an extension field of $E$), is a map which satisfies $\delta(x+y) ...
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366 views

Galois group of a non-separable polynomial

In my notes, I am given the definition of the Galois group of a polynomial only in the case when the polynomial is separable (if $f$ is a separable polynomial over $K$ with splitting field $L$, then ...
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Algebraic conjugates

Suppose $L/K$ is an algebraic field extension. Take $\alpha_1 \in L$. Then $\alpha_1$ has minimal polynomial $f(x)$ over $K$. Let $\alpha_2, ... \alpha_k$ be the other roots of $f$ in $L$. The ...
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The implications of the insolvability of certain polynomials

So I've just been over a bit of Galois theory, and I'm trying to understand what the implications are for a polynomial's Galois group to not be solvable. My book says this means that there is no ...
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Basic example of extensions of residue fields.

Can anyone think of a simple example of the following: $B/A$ is an integral extension of DVRs with quotient fields $L$ and $K$ and residue fields $\bar{L}$ and $\bar{K}$, $L/K$ is finite dimensional ...
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339 views

Product in GF(16)

i need some help with a product in GF(16), where it is seen as an extension of GF(4)={0, 1, x, x+1} (where $x^2 = x + 1$ ) with the irreducible polynom $f(y) = y^2 + y + x$ So the elements in the ...
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Splitting field implies Galois extension

I hope this isn't too elementary of a question, but I'm not sure I understand Artin's proof that if $K/F$ is a finite extension, then $K/F$ Galois is equivalent to $K$ being a splitting field over $F$ ...
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“The Galois group of $\pi$ is $\mathbb{Z}$”

Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question: The Galois group of $\pi$ is $\mathbb{Z}$. In what sense/framework is ...
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170 views

Counting endomorphisms of $\mathbf Q(\zeta _{n})$

If $\zeta= \zeta_{n}$, how does one count the homomorphisms $f:\mathbf{Q}(\zeta)\rightarrow \mathbf{Q}(\zeta)$?
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For $n\ge 3, x_{1},…,x_{n} \in \mathbf{Q}^{\ast}$, $[\mathbf{Q}(\sqrt{x_{1}},…\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$

For $n\ge 3, x_{1},...,x_{n} \in \mathbf{Q}^{\ast}$ and $[\mathbf{Q}(\sqrt{x_{1}},...\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$ how can we conclude that there are non empty $I \subset \{1,...,n\}$ with ...
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Can a regular heptagon be constructed using a compass, straightedge, and angle trisector?

Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
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How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots \sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?

This is Exercise 18.14 from Algebra, Isaacs. $p_{1}\ ,\ p_{2}\ ,\ ... p_{n}$ are different prime numbers. How to show that $$\mathbb{Q}[\sqrt{p_{1}}, \sqrt{p_{2}}, \ldots \sqrt{p_{n}} ] = ...